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Numberplay: Don’t Make a Triangle, Part 2

By Gary Antonick August 22, 2011 11:11 amAugust 22, 2011 11:11 am

Gary AntonickAnd that’s what a draw looks like.

Katherine Cook of the Math for Love team continues this week with the second in our two-part triangle series. After the puzzle you’ll find Ms. Cook’s essay on the importance of making mistakes in mathematics. Mistakes! Ms. Cook has good things to say about them. We’ve seen this in the best Numberplay discussions — the most productive inquiries come from what turn out to be initial mis-steps. So let’s go for an abundance of mistakes this week.

Here’s Ms. Cook:

We’re keeping with our theme of avoiding triangles, so let’s call today Avoid Triangles At All Costs (ATAAC) Day Two, and in honor of our second triangle-free day, we have the following celebratory puzzle. It’s called Don’t Make A Triangle II. Two players begin by drawing a number of dots in a circle. Let’s call the players Red and Blue. Red and Blue alternate connecting any two dots that have not yet been connected. The ﬁrst player to make a triangle on the dots with sides entirely of that player’s color loses.

The four-dot game. Red goes first. Blue second. Red third. Blue fourth. Red narrowly avoids a triangle in the fifth move. Blue is forced to make a triangle in the last move and loses. Red wins!

The three-dot game above is a draw, while the four-dot game is not.

Q1: Can you find a four-dot game that will end in a draw? What about a five-dot game?

Q2: Is there a number of dots that does not end in a draw no matter how Red and Blue play the game?

Solution

Here’s Katherine Cook with the official solution:

Great work this week, everyone.

Q1. Both the four-dot game and the five-dot game have draw options. An interesting further question about this is just how many different ways there are to draw in these games.

Q2. #16 Gary gave what is essentially the classic argument that with 6 dots, a draw is impossible. This type of argument requires sight of the implications inherent in a state of the game. For example, to start you say that your aim is to avoid monochromatic triangles. You know that by the end of the game, every dot will be connected to every other dot, so if you take Gary’s dot X, it has five dots it shares a connection with. Those connections are either red or blue. They could be all red, or all blue, or one red and four blue, or two red and three blue, and so on. But in any one of those cases, at least three of the connections are the same color. Let’s say those three are blue (does it matter which we choose?). Now it’s time to use the aim we started with: if we want to avoid monochromatic triangles, what has to be true about the connections between those dots on the other ends of the blue lines coming from X? Naming those dots will give us some flexibility in our description, so let’s take Gary’s A,B, and C. By the end of the game, A,B,C are all connected to each other, and none of the connections can be blue. If they were, say A to B was a blue line, then XAB would make a blue triangle. This means that A to B, B to C, and A to C are each a red connection, and we’ve now got a red triangle.

Last week I talked about how asking questions changes the power structure of mathematics. Questions transform math from something that is done to you to something that you do, and perhaps, something that you love.

Though there are many more reasons than I could share here that math is worth loving, one of the more salient is its irrepressible element of surprise. At almost every level, if you are willing to start the work of asking questions, you will find there are surprising and delightful things to discover, things you would not have expected to find. We should care about surprise; the element of surprise is the essence of learning, and is worth pursuing for as long as one lives.

Mathematics is a deep mine of surprise, and with patience and creativity it is yours to explore. For example, last week we worked on eliminating monochromatic equilateral triangles from an arrangement of dots. A rather reasonable first go on the problem left some readers with an incorrect, though not obviously so, count of the number of triangles. There are a few hidden in there, turned slightly, and thus harder to see. Error is an irrecusable fact of problem-solving, and this kind of error is definitely the rule and not the exception. Making this counting mistake has an added benefit, though: by stumbling down this wrong path, you’ll soon have the pleasure, the full surprise, of finding out there is a correct path. It’s just not the path you’re on at the moment. It’s like one of those computer-generated pictures that look like a garbled pattern at first but as you shift your focus, an image pops out of the page at you. By making the mistaken first count of triangles, you are now open to the surprise when you notice there are more triangles there than you initially thought. And for those of you who have ever had this kind of miscount happen, perhaps you remember the sense of surprise that accompanied the discovering of more there than you had thought.

In puzzle-solving especially, mistakes are a critical part of the process and also some of the richest, most fertile media for growing a sense of surprise. Surprise essentially requires mistakes: you have to have formed an opinion in order for it to be changed. So take heart in your errors, because surprise comes with being wrong, a vital event in mathematics. (One commenter on last week’s puzzle even charmingly mentioned that “every comment I have made so far contains an error,” a refreshing admission in a field that is often — and problematically — taught as an obsessive pursuit of right answers.) To err is human. In mathematics, error is also a sign that surprises may be close.

Another surprise we often see in mathematics: naturally distinct fields of mathematics can harmonize with each other. My background as a mathematician is in probabilistic combinatorics, the magical blend of probability and combinatorics, which deals with how to count things that have random behavior. The puzzle this week has roots in combinatorics and graph theory and even reaches into some very good stuff in probabilistic combinatorics. Stay tuned to the comments later in the week when I’ll explain how the puzzle relates to these surprising and appealing results in mathematics.

In honor of the spirit of surprise, let’s avoid revealing the solution for the first 24 hours the puzzle is up, and limit comments to lines of inquiry, questions, hints or observations. After 24 hours, full solutions are welcome and encouraged.

Notes

Ms. Cook has a master’s degree in mathematics from the University of Washington and designs and teaches enrichment courses in science and math in Seattle.

If you have a drawing you would like to append to your comment, please send to numberplay@nytimes.com.

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Welcome to our conversation about word games. Here you'll find a new blog post for each day's crossword plus a bonus post for the Variety puzzle. Along with discussion about the day's challenge, you'll get backstage insights about puzzlemaking and occasional notes from The Times's puzzlemaster, Will Shortz.

Deb Amlen is a humorist and puzzle constructor whose work has appeared in The New York Times, The Washington Post, The Los Angeles Times, The Onion and Bust Magazine. Her books, “It's Not P.M.S., It's You” and “Create Your Life Lists” are available where all fine literature is sold.

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About Numberplay, the Puzzle Suite For Math Lovers of All Ages

Numberplay is a puzzle suite that will be presented in Wordplay every Monday. The puzzles, which are inspired by many sources and are reported by Gary Antonick, are generally mathematical or logical problems, with occasional forays into physics and other branches of science. While written for adults, many of the concepts here are suitable for and can be enjoyed by math students of all ages.

Gary Antonick, who has created or edited over 100 logic and math puzzles for The New York Times, secretly believes every math problem can be solved using circles and straight lines. He is a visiting scholar at Stanford University, where he studies mathematical problem solving.

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